We now claim $\chi(B_i) \leq O( \frac{d}{\log d})$. To see that this is true, we suppose that each vertex has an available color palette of size $c \frac{d}{\log d}$ where $c$ is a sufficiently large constant.

Start at the largest value $j s+i$, and color $G[A_{j s + i}]$ using $O(d/\log d)$ colors (it has degree $d$ and is triangle free.)
Now look at the graph $G[A_{(j-1) s + i}]$. Each vertex touches at most $d 2^{-s} = d/\log d$ already-colored vertices. So it has at least $\Omega(\frac{d}{\log d})$ colors remaining in its palette. So it can be list-colored.

$\begingroup$Maybe, even the following is true? If the graph does not have triangles and has $O(d\cdot |V_1|)$ edges inbetween any set $V_1\subset V$, then $\chi(G)=O(d/\log d)$? (Equivalent requirement is that the vertices of $G$ may be ordered so that each vertex has at most $O(d)$ vertices with greater number assigned. This is what we get if order each $A_i$ arbitrarily.)$\endgroup$
– Fedor PetrovNov 18 '16 at 13:34

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$\begingroup$@FedorPetrov Your assumption is equivalent to graph having degeneracy $d$. Unfortunately, Johansson's theorem on chromatic number of triangle-free graphs fails for $d$-degenerate graphs. An example can be found in the concluding remarks of "Coloring graphs with sparse neighborhoods" by Alon, Krivelevich and Sudakov.$\endgroup$
– Boris BukhNov 21 '16 at 20:59